Lewandowski, 24, will move to the Bundesliga champions at the end of the season when his contract expires.
In an interview with Sport 1, Lewandowski indicated he should ink a deal in early 2014.
Asked about the possibility of concluding a transfer to Bayern in the new year, Lewandowski replied: "Yes, because then I can officially sign a contract.
"Officially, I can only say something in January. At this moment it is still too early. We still have to wait a bit."
One of the worst-kept secrets in German football, it means the Pole will spearhead Pep Guardiola's attack next season and be reunited with Dortmund teammate Mario Götze.
It is another hammer blow for Borussia, who would have preferred to sell Lewandowski abroad but will now lose another one of their key attackers to their main rivals in the top flight.
Götze left for Bayern after the Champions League winners triggered his €37 million release clause and though they were also prepared to bid for Lewandowski this summer, Borussia refused to sell another of their players to their competitor.
Instead they will lose Lewandowski for nothing, though Jürgen Klopp prepared for his inevitable departure with the signings of Henrikh Mkhitaryan and Pierre-Emerick Aubameyang.